On Some Linear Codes

TL;DR

This paper provides an alternative proof of Massey's theorem for linear codes with complementary duals, establishes bounds on the maximum minimum distance for ternary LCD codes, and discusses cases of bound attainment.

Contribution

It offers a new proof of Massey's theorem and derives bounds for the maximum minimum distance of ternary LCD codes, including conditions for when bounds are achieved.

Findings

Alternative proof of Massey's theorem for LCD codes

Bounds on maximum minimum distance of ternary LCD codes

Characterization of cases when bounds are attained

Abstract

Linear code with complementary dual($LCD$) are those codes which meet their duals trivially. In this paper we will give rather alternative proof of Massey's theorem\cite{Massey2}, which is one of the most important characterization of $LCD$ codes. Let $LCD[n,k]_3$ denote the maximum of possible values of $d$ among $[n,k,d]$ ternary codes. We will give bound on $LCD[n,k]_3$. We will also discuss the cases when this bound is attained.